In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.

Spectrum estimation and harmonic analysis

A study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and/or undamped sinusoids in noise is presented. Comparison of this method to a polynomial method (SVD-Prony method) shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes. It is found through perturbation analysis and simulation that, for signals with unknown damping factors, the pencil method is less sensitive to noise than the polynomial method. An expression of the Cramer-Rao bound for the exponential signals is presented. >

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Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise

We introduce key concepts and issues including the effects of atmospheric propagation upon the data, spectral variability, mixed pixels, and the distinction between classification and detection algorithms. Detection algorithms for full pixel targets are developed using the likelihood ratio approach. Subpixel target detection, which is more challenging due to background interference, is pursued using both statistical and subspace models for the description of spectral variability. Finally, we provide some results which illustrate the performance of some detection algorithms using real hyperspectral imaging (HSI) data. Furthermore, we illustrate the potential deviation of HSI data from normality and point to some distributions that may serve in the development of algorithms with better or more robust performance. We therefore focus on detection algorithms that assume multivariate normal distribution models for HSI data.

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Detection algorithms for hyperspectral imaging applications

The approximation of a function by a sum of complex exponentials is a problem that is at least two centuries old. Fundamentally, all techniques discussed in this article proceed from using the same sequence of data samples and vary only, but importantly, in how those samples are used in achieving the parameter estimation. All of these techniques, in other words, seek the same quantitative parameters to represent the sampled data, but use different routes to get there. The techniques for estimating the parameters are either linear or nonlinear. The linear techniques are emphasized in this presentation. In particular, the matrix pencil method is described, which is more robust to noise in the sampled data. The matrix pencil approach has a lower variance of the estimates of the parameters of interest than a polynomial-type method (Prony's method belongs to this category), and is also computationally more efficient. A bandpass version of the matrix pencil can be implemented in hardware, utilizing an AT&T DSP32C chip operating in real time. A copy of the computer program implementing the matrix pencil technique is given in the appendix. >

/pdf/using-the-matrix-pencil-method-to-estimate-the-parameters-of-4t6chz0vtb.pdf

Using the matrix pencil method to estimate the parameters of a sum of complex exponentials

Most simple nonlinear thresholding rules for wavelet-based denoising assume that the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependencies. We only consider the dependencies between the coefficients and their parents in detail. For this purpose, new non-Gaussian bivariate distributions are proposed, and corresponding nonlinear threshold functions (shrinkage functions) are derived from the models using Bayesian estimation theory. The new shrinkage functions do not assume the independence of wavelet coefficients. We show three image denoising examples in order to show the performance of these new bivariate shrinkage rules. In the second example, a simple subband-dependent data-driven image denoising system is described and compared with effective data-driven techniques in the literature, namely VisuShrink, SureShrink, BayesShrink, and hidden Markov models. In the third example, the same idea is applied to the dual-tree complex wavelet coefficients.

Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency

With the modeling of non-Gaussian radar clutter in mind, elegant and tractable techniques are presented for characterizing the probability density function (PDF) of a correlated non-Gaussian radar vector. The need for a library of multivariable correlated non-Gaussian PDFs in order to characterize various clutter scenarios is discussed. Specifically,. the theory of spherically invariant random processes (SIRPs) is examined in detail. Approaches based on the marginal envelope PDF and the marginal characteristic function have been used to obtain several multivariate non-Gaussian PDFs. An important result providing the PDF of the quadratic form of a spherically invariant random vector (SIRV) is presented. This result enables the problem of distributed identification of a SIRV to be addressed. >

Non-Gaussian random vector identification using spherically invariant random processes

A new framework for the ambiguity function for a bistatic radar is presented. Modeling of radar measurements of a slowly fluctuating point target involving delay and Doppler shift is considered. Effects of bistatic geometry on these measurements are discussed in detail. It is shown by means of two examples that geometry factors play an important role in the shape of the ambiguity function for a bistatic radar as the system configuration is varied.

Ambiguity function for a bistatic radar

We develop computer simulation procedures which enable us to generate any correlated non-Gaussian radar clutter that can be modeled as a spherically invariant random process (SIRP). In most cases, when the clutter is a correlated non-Gaussian random process, performance of the optimal radar signal processor cannot be evaluated analytically. Therefore, in order to evaluate such processors, there is a need for efficient computer simulation of the clutter. We present two canonical simulation procedures for the generation of correlated non-Gaussian clutter. A new approach for the goodness-of-fit test is proposed in order to assess the performance of the simulation procedure. >

Computer generation of correlated non-Gaussian radar clutter

Many problems of mathematical physics can be formulated in terms of the operator equation Ax = y , where A is an integro-differential operator. Given A and x , the solution for y is usually straightforward. However, the inverse problem which consists of the solution for x when given A and y is much more difficult. The following questions relative to the inverse problem are explored. 1) Does specification of the operator A determine the set \{y\} for which a solution x is possible? 2) Does the inverse problem always have a unique solution? 3) Do small perturbations of the forcing function y always result in small perturbations of the solution? 4) What are some of the considerations that enter into the choice of a solution technique for a specific problem? The concept of an ill-posed problem versus that of a well-posed problem is discussed. Specifically, the manner by which an ill-posed problem may be regularized to a well-posed problem is presented. The concepts are illustrated by several examples.

Some mathematical considerations in dealing with the inverse problem

An analysis of the ambiguity function for bistatic radar is presented. Radar measurements of a moving target are considered. The effect of bistatic geometry is discussed in detail. An example is provided to illustrate the nature of ambiguity function for a variety of system configurations. >

Donald D. Weiner

Papers

Non-Gaussian random vector identification using spherically invariant random processes

Ambiguity function for a bistatic radar

Computer generation of correlated non-Gaussian radar clutter

Some mathematical considerations in dealing with the inverse problem

Ambiguity function for a bistatic radar